A332742 Number of non-unimodal negated permutations of a multiset whose multiplicities are the prime indices of n.

0, 0, 0, 0, 0, 1, 0, 2, 3, 2, 0, 8, 0, 3, 7, 16, 0, 24, 0, 16, 12, 4, 0, 52, 16, 5, 81, 26, 0, 54, 0, 104, 18, 6, 31, 168, 0, 7, 25, 112, 0, 99, 0, 38, 201, 8, 0, 344, 65, 132, 33, 52, 0, 612, 52, 202, 42, 9, 0, 408, 0, 10, 411, 688, 80, 162, 0, 68, 52, 272

Offset

1,8

Comments

This multiset is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

A sequence of integers is unimodal if it is the concatenation of a weakly increasing and a weakly decreasing sequence.

Links

Table of n, a(n) for n=1..70.

Eric Weisstein's World of Mathematics, Unimodal Sequence

Formula

a(n) + A332741(n) = A318762(n).

Example

The a(n) permutations for n = 6, 8, 9, 10, 12, 14, 15, 16:
  121  132  1212  1121  1132  11121  11212  1243
       231  1221  1211  1213  11211  11221  1324
            2121        1231  12111  12112  1342
                        1312         12121  1423
                        1321         12211  1432
                        2131         21121  2143
                        2311         21211  2314
                        3121                2341
                                            2413
                                            2431
                                            3142
                                            3241
                                            3412
                                            3421
                                            4132
                                            4231

Mathematica

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
unimodQ[q_]:=Or[Length[q]<=1, If[q[[1]]<=q[[2]], unimodQ[Rest[q]], OrderedQ[Reverse[q]]]];
Table[Length[Select[Permutations[nrmptn[n]], !unimodQ[#]&]], {n, 30}]

Crossrefs

Dominated by A318762.

The complement of the non-negated version is counted by A332294.

The non-negated version is A332672.

The complement is counted by A332741.

A less interesting version is A333146.

Unimodal compositions are A001523.

Unimodal normal sequences are A007052.

Non-unimodal normal sequences are A328509.

Partitions with non-unimodal 0-appended first differences are A332284.

Compositions whose negation is unimodal are A332578.

Partitions with non-unimodal negated run-lengths are A332639.

Numbers whose negated prime signature is not unimodal are A332642.

Cf. A056239, A112798, A115981, A124010, A181819, A181821, A304660, A332280, A332283, A332288, A332638, A332669, A333145.

Sequence in context: A105855 A152954 A079175 * A202815 A049336 A017888

Adjacent sequences: A332739 A332740 A332741 * A332743 A332744 A332745

Keyword

nonn

Author

Gus Wiseman, Mar 09 2020

Status

approved